I think that the easiest example of compact pseudo-Kähler manifold which does not admit any Kähler metric is the [Kodaira-Thurston manifold][1]. See for instance the introduction of

<cite authors="Yamada, Takumi">_Yamada, Takumi_, [**Ricci flatness of certain compact pseudo-Kähler solvmanifolds**](https://doi.org/10.1016/j.geomphys.2011.06.006), J. Geom. Phys. 62, No. 5, 1338-1345 (2012). [ZBL1239.53100](https://zbmath.org/?q=an:1239.53100).</cite>


  [1]: https://www.jstor.org/stable/2041749?origin=crossref#metadata_info_tab_contents